The 1st, 5th, and 13th terms of an arithmetic sequence are the first three terms of a geometric sequence with a common ratio of 2. If the 21st term of the arithmetic sequence is 72, calculate the sum of the first 10 terms of the geometric sequence.

Question

here_to_help15 · Answer

the 1st term of the A.S. is a, and the 5th term is a + 4k 

from the G.S. with a common ratio of 2 
(a + 4k)/a = 2 
a + 4k = 2a 
a = 4k 

then from the 21st term of the A.S. 
a + 20k = 72 
substitute a = 4k 
4k + 20k = 72 
24k = 72 
k = 3 

a = 4*3 
a = 12 

the sum of the first 10 terms of the G.S. is 
S_10 = 12*(1 - 2^10)/(1 - 2) 
S_10 = 12276